# American Institute of Mathematical Sciences

September  2007, 6(3): 809-818. doi: 10.3934/cpaa.2007.6.809

## Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions

Received  April 2006 Revised  January 2007 Published  June 2007

Generalized Routh-Hurwitz conditions consist of the positivity of $n$ determinants associated to a polynomial of degree $n$. They can be used in order to guarantee that a refinable function with dilation $M$ is a ripplet, that is, the collocation matrices of its shifts are totally positive. Given a polynomial of degree $n$, a test of $\mathcal O(n^2)$ elementary operations and growth factor 1 is presented in order to check the generalized Routh-Hurwitz conditions. The case corresponding to $M=3$ is described in detail.
Citation: J. M. Peña. Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions. Communications on Pure & Applied Analysis, 2007, 6 (3) : 809-818. doi: 10.3934/cpaa.2007.6.809
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